MAYBE
851.52/297.04	MAYBE
851.52/297.04	
851.52/297.04	We are left with following problem, upon which TcT provides the
851.52/297.04	certificate MAYBE.
851.52/297.04	
851.52/297.04	Strict Trs:
851.52/297.04	  { eq(0(), 0()) -> true()
851.52/297.04	  , eq(0(), s(x)) -> false()
851.52/297.04	  , eq(s(x), 0()) -> false()
851.52/297.04	  , eq(s(x), s(y)) -> eq(x, y)
851.52/297.04	  , le(0(), y) -> true()
851.52/297.04	  , le(s(x), 0()) -> false()
851.52/297.04	  , le(s(x), s(y)) -> le(x, y)
851.52/297.04	  , app(nil(), y) -> y
851.52/297.04	  , app(add(n, x), y) -> add(n, app(x, y))
851.52/297.04	  , min(add(n, nil())) -> n
851.52/297.04	  , min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
851.52/297.04	  , if_min(true(), add(n, add(m, x))) -> min(add(n, x))
851.52/297.04	  , if_min(false(), add(n, add(m, x))) -> min(add(m, x))
851.52/297.04	  , head(add(n, x)) -> n
851.52/297.04	  , tail(nil()) -> nil()
851.52/297.04	  , tail(add(n, x)) -> x
851.52/297.04	  , null(nil()) -> true()
851.52/297.04	  , null(add(n, x)) -> false()
851.52/297.04	  , rm(n, nil()) -> nil()
851.52/297.04	  , rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
851.52/297.04	  , if_rm(true(), n, add(m, x)) -> rm(n, x)
851.52/297.04	  , if_rm(false(), n, add(m, x)) -> add(m, rm(n, x))
851.52/297.04	  , minsort(x) -> mins(x, nil(), nil())
851.52/297.04	  , mins(x, y, z) -> if(null(x), x, y, z)
851.52/297.04	  , if(true(), x, y, z) -> z
851.52/297.04	  , if(false(), x, y, z) -> if2(eq(head(x), min(x)), x, y, z)
851.52/297.04	  , if2(true(), x, y, z) ->
851.52/297.04	    mins(app(rm(head(x), tail(x)), y),
851.52/297.04	         nil(),
851.52/297.04	         app(z, add(head(x), nil())))
851.52/297.04	  , if2(false(), x, y, z) -> mins(tail(x), add(head(x), y), z) }
851.52/297.04	Obligation:
851.52/297.04	  runtime complexity
851.52/297.04	Answer:
851.52/297.04	  MAYBE
851.52/297.04	
851.52/297.04	None of the processors succeeded.
851.52/297.04	
851.52/297.04	Details of failed attempt(s):
851.52/297.04	-----------------------------
851.52/297.04	1) 'With Problem ... (timeout of 297 seconds)' failed due to the
851.52/297.04	   following reason:
851.52/297.04	   
851.52/297.04	   Computation stopped due to timeout after 297.0 seconds.
851.52/297.04	
851.52/297.04	2) 'Best' failed due to the following reason:
851.52/297.04	   
851.52/297.04	   None of the processors succeeded.
851.52/297.04	   
851.52/297.04	   Details of failed attempt(s):
851.52/297.04	   -----------------------------
851.52/297.04	   1) 'With Problem ... (timeout of 148 seconds) (timeout of 297
851.52/297.04	      seconds)' failed due to the following reason:
851.52/297.04	      
851.52/297.04	      Computation stopped due to timeout after 148.0 seconds.
851.52/297.04	   
851.52/297.04	   2) 'Best' failed due to the following reason:
851.52/297.04	      
851.52/297.04	      None of the processors succeeded.
851.52/297.04	      
851.52/297.04	      Details of failed attempt(s):
851.52/297.04	      -----------------------------
851.52/297.04	      1) 'bsearch-popstar (timeout of 297 seconds)' failed due to the
851.52/297.04	         following reason:
851.52/297.04	         
851.52/297.04	         The processor is inapplicable, reason:
851.52/297.04	           Processor only applicable for innermost runtime complexity analysis
851.52/297.04	      
851.52/297.04	      2) 'Polynomial Path Order (PS) (timeout of 297 seconds)' failed due
851.52/297.04	         to the following reason:
851.52/297.04	         
851.52/297.04	         The processor is inapplicable, reason:
851.52/297.04	           Processor only applicable for innermost runtime complexity analysis
851.52/297.04	      
851.52/297.04	   
851.52/297.04	   3) 'Fastest (timeout of 24 seconds) (timeout of 297 seconds)'
851.52/297.04	      failed due to the following reason:
851.52/297.04	      
851.52/297.04	      None of the processors succeeded.
851.52/297.04	      
851.52/297.04	      Details of failed attempt(s):
851.52/297.04	      -----------------------------
851.52/297.04	      1) 'Bounds with minimal-enrichment and initial automaton 'match''
851.52/297.04	         failed due to the following reason:
851.52/297.04	         
851.52/297.04	         match-boundness of the problem could not be verified.
851.52/297.04	      
851.52/297.04	      2) 'Bounds with perSymbol-enrichment and initial automaton 'match''
851.52/297.04	         failed due to the following reason:
851.52/297.04	         
851.52/297.04	         match-boundness of the problem could not be verified.
851.52/297.04	      
851.52/297.04	   
851.52/297.04	
851.52/297.04	3) 'Weak Dependency Pairs (timeout of 297 seconds)' failed due to
851.52/297.04	   the following reason:
851.52/297.04	   
851.52/297.04	   We add the following weak dependency pairs:
851.52/297.04	   
851.52/297.04	   Strict DPs:
851.52/297.04	     { eq^#(0(), 0()) -> c_1()
851.52/297.04	     , eq^#(0(), s(x)) -> c_2()
851.52/297.04	     , eq^#(s(x), 0()) -> c_3()
851.52/297.04	     , eq^#(s(x), s(y)) -> c_4(eq^#(x, y))
851.52/297.04	     , le^#(0(), y) -> c_5()
851.52/297.04	     , le^#(s(x), 0()) -> c_6()
851.52/297.04	     , le^#(s(x), s(y)) -> c_7(le^#(x, y))
851.52/297.04	     , app^#(nil(), y) -> c_8(y)
851.52/297.04	     , app^#(add(n, x), y) -> c_9(n, app^#(x, y))
851.52/297.04	     , min^#(add(n, nil())) -> c_10(n)
851.52/297.04	     , min^#(add(n, add(m, x))) ->
851.52/297.04	       c_11(if_min^#(le(n, m), add(n, add(m, x))))
851.52/297.04	     , if_min^#(true(), add(n, add(m, x))) -> c_12(min^#(add(n, x)))
851.52/297.04	     , if_min^#(false(), add(n, add(m, x))) -> c_13(min^#(add(m, x)))
851.52/297.04	     , head^#(add(n, x)) -> c_14(n)
851.52/297.04	     , tail^#(nil()) -> c_15()
851.52/297.04	     , tail^#(add(n, x)) -> c_16(x)
851.52/297.04	     , null^#(nil()) -> c_17()
851.52/297.04	     , null^#(add(n, x)) -> c_18()
851.52/297.04	     , rm^#(n, nil()) -> c_19()
851.52/297.04	     , rm^#(n, add(m, x)) -> c_20(if_rm^#(eq(n, m), n, add(m, x)))
851.52/297.04	     , if_rm^#(true(), n, add(m, x)) -> c_21(rm^#(n, x))
851.52/297.04	     , if_rm^#(false(), n, add(m, x)) -> c_22(m, rm^#(n, x))
851.52/297.04	     , minsort^#(x) -> c_23(mins^#(x, nil(), nil()))
851.52/297.04	     , mins^#(x, y, z) -> c_24(if^#(null(x), x, y, z))
851.52/297.04	     , if^#(true(), x, y, z) -> c_25(z)
851.52/297.04	     , if^#(false(), x, y, z) ->
851.52/297.04	       c_26(if2^#(eq(head(x), min(x)), x, y, z))
851.52/297.04	     , if2^#(true(), x, y, z) ->
851.52/297.04	       c_27(mins^#(app(rm(head(x), tail(x)), y),
851.52/297.04	                   nil(),
851.52/297.04	                   app(z, add(head(x), nil()))))
851.52/297.04	     , if2^#(false(), x, y, z) ->
851.52/297.04	       c_28(mins^#(tail(x), add(head(x), y), z)) }
851.52/297.04	   
851.52/297.04	   and mark the set of starting terms.
851.52/297.04	   
851.52/297.04	   We are left with following problem, upon which TcT provides the
851.52/297.04	   certificate MAYBE.
851.52/297.04	   
851.52/297.04	   Strict DPs:
851.52/297.04	     { eq^#(0(), 0()) -> c_1()
851.52/297.04	     , eq^#(0(), s(x)) -> c_2()
851.52/297.04	     , eq^#(s(x), 0()) -> c_3()
851.52/297.04	     , eq^#(s(x), s(y)) -> c_4(eq^#(x, y))
851.52/297.04	     , le^#(0(), y) -> c_5()
851.52/297.04	     , le^#(s(x), 0()) -> c_6()
851.52/297.04	     , le^#(s(x), s(y)) -> c_7(le^#(x, y))
851.52/297.04	     , app^#(nil(), y) -> c_8(y)
851.52/297.04	     , app^#(add(n, x), y) -> c_9(n, app^#(x, y))
851.52/297.04	     , min^#(add(n, nil())) -> c_10(n)
851.52/297.04	     , min^#(add(n, add(m, x))) ->
851.52/297.04	       c_11(if_min^#(le(n, m), add(n, add(m, x))))
851.52/297.04	     , if_min^#(true(), add(n, add(m, x))) -> c_12(min^#(add(n, x)))
851.52/297.04	     , if_min^#(false(), add(n, add(m, x))) -> c_13(min^#(add(m, x)))
851.52/297.04	     , head^#(add(n, x)) -> c_14(n)
851.52/297.04	     , tail^#(nil()) -> c_15()
851.52/297.04	     , tail^#(add(n, x)) -> c_16(x)
851.52/297.04	     , null^#(nil()) -> c_17()
851.52/297.04	     , null^#(add(n, x)) -> c_18()
851.52/297.04	     , rm^#(n, nil()) -> c_19()
851.52/297.04	     , rm^#(n, add(m, x)) -> c_20(if_rm^#(eq(n, m), n, add(m, x)))
851.52/297.04	     , if_rm^#(true(), n, add(m, x)) -> c_21(rm^#(n, x))
851.52/297.04	     , if_rm^#(false(), n, add(m, x)) -> c_22(m, rm^#(n, x))
851.52/297.04	     , minsort^#(x) -> c_23(mins^#(x, nil(), nil()))
851.52/297.04	     , mins^#(x, y, z) -> c_24(if^#(null(x), x, y, z))
851.52/297.04	     , if^#(true(), x, y, z) -> c_25(z)
851.52/297.04	     , if^#(false(), x, y, z) ->
851.52/297.04	       c_26(if2^#(eq(head(x), min(x)), x, y, z))
851.52/297.04	     , if2^#(true(), x, y, z) ->
851.52/297.04	       c_27(mins^#(app(rm(head(x), tail(x)), y),
851.52/297.04	                   nil(),
851.52/297.04	                   app(z, add(head(x), nil()))))
851.52/297.04	     , if2^#(false(), x, y, z) ->
851.52/297.04	       c_28(mins^#(tail(x), add(head(x), y), z)) }
851.52/297.04	   Strict Trs:
851.52/297.04	     { eq(0(), 0()) -> true()
851.52/297.04	     , eq(0(), s(x)) -> false()
851.52/297.04	     , eq(s(x), 0()) -> false()
851.52/297.04	     , eq(s(x), s(y)) -> eq(x, y)
851.52/297.04	     , le(0(), y) -> true()
851.52/297.04	     , le(s(x), 0()) -> false()
851.52/297.04	     , le(s(x), s(y)) -> le(x, y)
851.52/297.04	     , app(nil(), y) -> y
851.52/297.04	     , app(add(n, x), y) -> add(n, app(x, y))
851.52/297.04	     , min(add(n, nil())) -> n
851.52/297.04	     , min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
851.52/297.04	     , if_min(true(), add(n, add(m, x))) -> min(add(n, x))
851.52/297.04	     , if_min(false(), add(n, add(m, x))) -> min(add(m, x))
851.52/297.04	     , head(add(n, x)) -> n
851.52/297.04	     , tail(nil()) -> nil()
851.52/297.04	     , tail(add(n, x)) -> x
851.52/297.04	     , null(nil()) -> true()
851.52/297.04	     , null(add(n, x)) -> false()
851.52/297.04	     , rm(n, nil()) -> nil()
851.52/297.04	     , rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
851.52/297.04	     , if_rm(true(), n, add(m, x)) -> rm(n, x)
851.52/297.04	     , if_rm(false(), n, add(m, x)) -> add(m, rm(n, x))
851.52/297.04	     , minsort(x) -> mins(x, nil(), nil())
851.52/297.04	     , mins(x, y, z) -> if(null(x), x, y, z)
851.52/297.04	     , if(true(), x, y, z) -> z
851.52/297.04	     , if(false(), x, y, z) -> if2(eq(head(x), min(x)), x, y, z)
851.52/297.04	     , if2(true(), x, y, z) ->
851.52/297.04	       mins(app(rm(head(x), tail(x)), y),
851.52/297.04	            nil(),
851.52/297.04	            app(z, add(head(x), nil())))
851.52/297.04	     , if2(false(), x, y, z) -> mins(tail(x), add(head(x), y), z) }
851.52/297.04	   Obligation:
851.52/297.04	     runtime complexity
851.52/297.04	   Answer:
851.52/297.04	     MAYBE
851.52/297.04	   
851.52/297.04	   We estimate the number of application of {1,2,3,5,6,15,17,18,19} by
851.52/297.04	   applications of Pre({1,2,3,5,6,15,17,18,19}) =
851.52/297.04	   {4,7,8,9,10,14,16,21,22,25}. Here rules are labeled as follows:
851.52/297.04	   
851.52/297.04	     DPs:
851.52/297.04	       { 1: eq^#(0(), 0()) -> c_1()
851.52/297.04	       , 2: eq^#(0(), s(x)) -> c_2()
851.52/297.04	       , 3: eq^#(s(x), 0()) -> c_3()
851.52/297.04	       , 4: eq^#(s(x), s(y)) -> c_4(eq^#(x, y))
851.52/297.04	       , 5: le^#(0(), y) -> c_5()
851.52/297.04	       , 6: le^#(s(x), 0()) -> c_6()
851.52/297.04	       , 7: le^#(s(x), s(y)) -> c_7(le^#(x, y))
851.52/297.04	       , 8: app^#(nil(), y) -> c_8(y)
851.52/297.04	       , 9: app^#(add(n, x), y) -> c_9(n, app^#(x, y))
851.52/297.04	       , 10: min^#(add(n, nil())) -> c_10(n)
851.52/297.04	       , 11: min^#(add(n, add(m, x))) ->
851.52/297.04	             c_11(if_min^#(le(n, m), add(n, add(m, x))))
851.52/297.04	       , 12: if_min^#(true(), add(n, add(m, x))) -> c_12(min^#(add(n, x)))
851.52/297.04	       , 13: if_min^#(false(), add(n, add(m, x))) ->
851.52/297.04	             c_13(min^#(add(m, x)))
851.52/297.04	       , 14: head^#(add(n, x)) -> c_14(n)
851.52/297.04	       , 15: tail^#(nil()) -> c_15()
851.52/297.04	       , 16: tail^#(add(n, x)) -> c_16(x)
851.52/297.04	       , 17: null^#(nil()) -> c_17()
851.52/297.04	       , 18: null^#(add(n, x)) -> c_18()
851.52/297.04	       , 19: rm^#(n, nil()) -> c_19()
851.52/297.04	       , 20: rm^#(n, add(m, x)) -> c_20(if_rm^#(eq(n, m), n, add(m, x)))
851.52/297.04	       , 21: if_rm^#(true(), n, add(m, x)) -> c_21(rm^#(n, x))
851.52/297.04	       , 22: if_rm^#(false(), n, add(m, x)) -> c_22(m, rm^#(n, x))
851.52/297.04	       , 23: minsort^#(x) -> c_23(mins^#(x, nil(), nil()))
851.52/297.04	       , 24: mins^#(x, y, z) -> c_24(if^#(null(x), x, y, z))
851.52/297.04	       , 25: if^#(true(), x, y, z) -> c_25(z)
851.52/297.04	       , 26: if^#(false(), x, y, z) ->
851.52/297.04	             c_26(if2^#(eq(head(x), min(x)), x, y, z))
851.52/297.04	       , 27: if2^#(true(), x, y, z) ->
851.52/297.04	             c_27(mins^#(app(rm(head(x), tail(x)), y),
851.52/297.04	                         nil(),
851.52/297.04	                         app(z, add(head(x), nil()))))
851.52/297.04	       , 28: if2^#(false(), x, y, z) ->
851.52/297.04	             c_28(mins^#(tail(x), add(head(x), y), z)) }
851.52/297.04	   
851.52/297.04	   We are left with following problem, upon which TcT provides the
851.52/297.04	   certificate MAYBE.
851.52/297.04	   
851.52/297.04	   Strict DPs:
851.52/297.04	     { eq^#(s(x), s(y)) -> c_4(eq^#(x, y))
851.52/297.04	     , le^#(s(x), s(y)) -> c_7(le^#(x, y))
851.52/297.04	     , app^#(nil(), y) -> c_8(y)
851.52/297.04	     , app^#(add(n, x), y) -> c_9(n, app^#(x, y))
851.52/297.04	     , min^#(add(n, nil())) -> c_10(n)
851.52/297.04	     , min^#(add(n, add(m, x))) ->
851.52/297.04	       c_11(if_min^#(le(n, m), add(n, add(m, x))))
851.52/297.04	     , if_min^#(true(), add(n, add(m, x))) -> c_12(min^#(add(n, x)))
851.52/297.04	     , if_min^#(false(), add(n, add(m, x))) -> c_13(min^#(add(m, x)))
851.52/297.04	     , head^#(add(n, x)) -> c_14(n)
851.52/297.04	     , tail^#(add(n, x)) -> c_16(x)
851.52/297.04	     , rm^#(n, add(m, x)) -> c_20(if_rm^#(eq(n, m), n, add(m, x)))
851.52/297.04	     , if_rm^#(true(), n, add(m, x)) -> c_21(rm^#(n, x))
851.52/297.04	     , if_rm^#(false(), n, add(m, x)) -> c_22(m, rm^#(n, x))
851.52/297.04	     , minsort^#(x) -> c_23(mins^#(x, nil(), nil()))
851.52/297.04	     , mins^#(x, y, z) -> c_24(if^#(null(x), x, y, z))
851.52/297.04	     , if^#(true(), x, y, z) -> c_25(z)
851.52/297.04	     , if^#(false(), x, y, z) ->
851.52/297.04	       c_26(if2^#(eq(head(x), min(x)), x, y, z))
851.52/297.04	     , if2^#(true(), x, y, z) ->
851.52/297.04	       c_27(mins^#(app(rm(head(x), tail(x)), y),
851.52/297.04	                   nil(),
851.52/297.04	                   app(z, add(head(x), nil()))))
851.52/297.04	     , if2^#(false(), x, y, z) ->
851.52/297.04	       c_28(mins^#(tail(x), add(head(x), y), z)) }
851.52/297.04	   Strict Trs:
851.52/297.04	     { eq(0(), 0()) -> true()
851.52/297.04	     , eq(0(), s(x)) -> false()
851.52/297.04	     , eq(s(x), 0()) -> false()
851.52/297.04	     , eq(s(x), s(y)) -> eq(x, y)
851.52/297.04	     , le(0(), y) -> true()
851.52/297.04	     , le(s(x), 0()) -> false()
851.52/297.04	     , le(s(x), s(y)) -> le(x, y)
851.52/297.04	     , app(nil(), y) -> y
851.52/297.04	     , app(add(n, x), y) -> add(n, app(x, y))
851.52/297.04	     , min(add(n, nil())) -> n
851.52/297.04	     , min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
851.52/297.04	     , if_min(true(), add(n, add(m, x))) -> min(add(n, x))
851.52/297.04	     , if_min(false(), add(n, add(m, x))) -> min(add(m, x))
851.52/297.04	     , head(add(n, x)) -> n
851.52/297.04	     , tail(nil()) -> nil()
851.52/297.04	     , tail(add(n, x)) -> x
851.52/297.04	     , null(nil()) -> true()
851.52/297.04	     , null(add(n, x)) -> false()
851.52/297.04	     , rm(n, nil()) -> nil()
851.52/297.04	     , rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
851.52/297.04	     , if_rm(true(), n, add(m, x)) -> rm(n, x)
851.52/297.04	     , if_rm(false(), n, add(m, x)) -> add(m, rm(n, x))
851.52/297.04	     , minsort(x) -> mins(x, nil(), nil())
851.52/297.04	     , mins(x, y, z) -> if(null(x), x, y, z)
851.52/297.04	     , if(true(), x, y, z) -> z
851.52/297.04	     , if(false(), x, y, z) -> if2(eq(head(x), min(x)), x, y, z)
851.52/297.04	     , if2(true(), x, y, z) ->
851.52/297.04	       mins(app(rm(head(x), tail(x)), y),
851.52/297.04	            nil(),
851.52/297.04	            app(z, add(head(x), nil())))
851.52/297.04	     , if2(false(), x, y, z) -> mins(tail(x), add(head(x), y), z) }
851.52/297.04	   Weak DPs:
851.52/297.04	     { eq^#(0(), 0()) -> c_1()
851.52/297.04	     , eq^#(0(), s(x)) -> c_2()
851.52/297.04	     , eq^#(s(x), 0()) -> c_3()
851.52/297.04	     , le^#(0(), y) -> c_5()
851.52/297.04	     , le^#(s(x), 0()) -> c_6()
851.52/297.04	     , tail^#(nil()) -> c_15()
851.52/297.04	     , null^#(nil()) -> c_17()
851.52/297.04	     , null^#(add(n, x)) -> c_18()
851.52/297.04	     , rm^#(n, nil()) -> c_19() }
851.52/297.04	   Obligation:
851.52/297.04	     runtime complexity
851.52/297.04	   Answer:
851.52/297.04	     MAYBE
851.52/297.04	   
851.52/297.04	   Empty strict component of the problem is NOT empty.
851.52/297.04	
851.52/297.04	
851.52/297.04	Arrrr..
851.64/297.16	EOF